Optimization problem: how to limit number of transitions from 0

Hello everyone

I am developping a solution for an optimization problem whose output x is made of 24 variables, like follows.

min(x) f'*x s.t. min_p <=x(i) <= max_p (48 constraints)

Among the other constraints I should add a constraint (maybe other 24 constraints) which limit the maximum number of startups.

For a simple understanding of what I have to reach:

everytime the optimizer evaluates a solution where x(i) is 0 and x(i+1)>0, total_startups = total_startups +1;

and in the end total_startups <= max_startups.

I am starting using intlinprog and thinking about the idea to write some constraints where to use integer variables (BIM).

Can you suggest any idea?

Answer 1

[Sameer]

Hi Chiara

To model the constraint that limits the maximum number of startups in your optimization problem, you can indeed use binary integer variables and additional constraints. The idea is to introduce binary variables that capture the "startup" event, i.e., when your system transitions from an off state (x(i) = 0) to an on state (x(i+1) > 0). Let's denote these binary variables as y(i) for i = 1, ..., 23 (since a startup can only occur between these intervals in a 24-period model).

Here's how you can approach this:

Define Binary Variables: For each interval i where a startup can occur, define a binary variable y(i) that takes the value 1 if a startup occurs between x(i) and x(i+1).

Startup Detection Constraints: For each i = 1, ..., 23, you need constraints that force y(i) to be 1 if x(i) = 0 and x(i+1) > 0. This can be achieved by using big-M method constraints. You'll select a sufficiently large positive number M that is greater than any possible value of x(i+1) when x(i+1) > 0.

The constraints can be formulated as:

x(i+1) - M*y(i) <= 0 for i = 1, ..., 23

This ensures that if x(i+1) > 0, then y(i) must be 1 to satisfy the constraint, because M is large. If x(i+1) = 0, y(i) can be 0, satisfying the constraint without triggering a startup.

Limiting Total Startups: To ensure the total number of startups does not exceed max_startups, you can add the following constraint:

Additional Considerations: Ensure that your choice of M is large enough to not interfere with the feasibility of other constraints but not excessively large to cause numerical stability issues in the solver.

Implementation in intlinprog: Define your binary variables and add the constraints as described above. Remember to declare y(i) as binary variables in the solver.

I hope this helps!

Sameer